In the last step we used the fact that the integral in the \(x\)  variable evaluates to \(1/4\) .
This result is consistent with Eq. (REF ), obtained with the more commonly used integration variable \(k_E^2\) . The advantage of the standard approach is that the result directly applies to an arbitrary pion vector form factor, while collecting the residues in the \(k^0\)  integration assumes the form dictated by vector meson dominance. For our application this approximation is completely sufficient, but could be improved by using a dispersive representation of \(F_\pi ^V(k^2)\)  [1]}, [2]} and treating the Cauchy kernel along the same lines.
